What is the Limit of a Sequence with Infinite Terms?

[aid]

Homework Statement

So, I am to calculate limit of a sequence given by a formula:

$$\sum^{n}_{i = 1} \sum^{i}_{j = 1} \frac{j}{n^3}$$

The Attempt at a Solution

I've tried to write down the sequence explicite and this is what I get:

$$\frac{1}{n^3} + (\frac{1}{n^3} + \frac{2}{n^3}) + ... + (\frac{1}{n^3} + ... + \frac{n}{n^3})$$

The last, n-th element could be written as:

$$\frac{\frac{n(n+1)}{2}}{n^3}$$,

the n-1-th element as:

$$\frac{\frac{n(n+1)}{2}}{n^3} - \frac{n}{n^3}$$

and so on. In other words:

$$\sum^{n}_{i = 1} \sum^{i}_{j = 1} \frac{j}{n^3} = n \frac{n(n+1)}{2n^3} - (n - 1)\frac{n}{n^3} - (n - 2)\frac{n -1}{n^3} - (n -3)\frac{n - 2}{n^3} - ... - \frac{2}{n^3}$$

If $$n \rightarrow \infty$$ then the right side of the equation goes to $$\frac{1}{2}$$. But the right answer is $$\frac{1}{6}$$.

Could anybody tell me what the heck I'm doing wrong?

 

[Karlx]

Your last sentence was
"If n tends to infinity then the right side of the equation goes to 1/2".

Think carefully how are you calculating this limit.
You must take into account that when n tends to infinity, the right side of the equation has an infinite number of terms.
Before calculating the limit, better you try to write the (n-1) last terms in a compact expression.